## Structure and Definability in General Bounded Arithmetic Theories (1999)

Citations: | 23 - 7 self |

### BibTeX

@MISC{Pollett99structureand,

author = {Chris Pollett},

title = {Structure and Definability in General Bounded Arithmetic Theories},

year = {1999}

}

### OpenURL

### Abstract

This paper is motivated by the questions: what are the \Sigma

### Citations

235 | Almost optimal lower bounds for small depth circuits
- H˚astad
- 1989
(Show Context)
Citation Context ...1, k ≥ 0) Let ℓ ∈ τ. The circuit P ℓ,k i P ℓ i (k, α) under the assignment pv,y1,...,yi = ⎧ 1 if 〈k, v, y1, . . . , yi〉 ∈ α ⎪⎨ ⎪⎩ 0 otherwise The next definition is needed to apply a result of Hastad =-=[13]-=-. yi=0 is computes the value of Definition 85 (i) Let (Bj)j be a partition of the atoms of P ℓ,k i into 2 ||ℓ(k)||2 · k i−1 classes of the form � � � pv,y1,...,yi−1,yi �yi < �i · k log(k) �1/2� 2 one ... |

231 |
Bounded Arithmetic, Propositional Logic, and Complexity Theory, volume 60 of Encyclopedia of Mathematics and Its Applications
- Kraj́ıček
- 1995
(Show Context)
Citation Context ...x, v, α) ∧ v = 1mod2 ∧ ¬(∃v ′ < 2 ||ℓ(x)||2 )(v ′ > v ∧ Ψi(x, v ′ , α))] P ℓ i (x, α) is true if the maximal v satisfying Ψi(x, v, α) is odd. We have modified the definition of Ψi above from Krajíček =-=[18]-=- and so have not entirely directly adapted the problem used there to separate S i 2(α) from T i 2(α). We did this to simplify our proof of Lemma 86 and because Lemma 91 seemed harder to show using a m... |

123 |
Bounded Arithmetic, Bibliopolis
- Buss
- 1986
(Show Context)
Citation Context ..., multivalued functions, conservation results, oracle separations 1991 MSC: 03F30, 68Q15 1 Introduction Three families of bounded arithmetic theories, R i 2 , S i 2, and T i 2, were developed in Buss =-=[7]-=-, Allen [1],Clote-Takeuti [11], and Takeuti [28]. These theories Preprint submitted to Elsevier Preprint 20 February 2003shave been studied because of their close connection to computational complexit... |

118 |
Feasibly constructive proofs and the propositional calculus
- Cook
- 1975
(Show Context)
Citation Context ...quivalent to a formula of a particular syntactic form. As an application of this we give a proof theoretic proof that Si 2 admits ∆b i+1induction. This solves open question (10) of Clote and Krajíček =-=[12]-=-. Further restricting this syntactic characterisation might be helpful in the development of separation results. The last section of this paper gives some oracle separations based on our syntactic cha... |

117 |
Proof theory
- TAKEUTI
- 1987
(Show Context)
Citation Context ... ˆT i,τ 2 and Ĉi,τ 2 proofs. To show the converse of Theorem 35; however, we work in the sequent calculus reformulating the induction and replacement axioms as rules of inference. Buss [7] or Takeuti =-=[27]-=- describe the sequent calculus. Definition 41 A Ψ-IND τ inference is an inference: A(b), Γ → A(Sb), ∆ A(0), Γ → A(ℓ(t(x))), ∆ where b is an eigenvariable and must not appear in the lower sequent, t ∈ ... |

87 |
Existence and Feasibility in Arithmetic
- Parikh
- 1971
(Show Context)
Citation Context ... L ˆ Σ b i-formulas provable in ˆ T i,τ 2 has a ˆ T i,τ 2 -proof in which only L ˆ Σ b i-formulas occur. 4.1 The witness predicate Let T be one of EBASIC2, ˆ T i,τ 2 , or Ĉi,τ 2 . By Parikh’s Theorem =-=[21]-=-, T can ˆΣ b m-define a function f if and only if there is a ˆ Σb m-formula Af(x, y) and a term t ∈ L2 such that T proves (∀x)(∃!y ≤ t)Af(x, y). For a multifunction one does not have to show uniquenes... |

71 |
The Polynomial Time Hierarchy Collapses if the Boolean Hierarchy Collapses
- Kadin
- 1988
(Show Context)
Citation Context ...ot separate these theories unless the polynomial hierarchy is infinite. The results of Hemaspaandra, Hemaspaandra, Hempel [14,15] and Buhrman Fortnow [4] are based on the easy-hard arguments of Kadin =-=[16]-=- and are of a simplistic enough nature that they might be formalizable in T i 2. This would give a provable collapse to B(Σ p i+2) if T i 2 = ˆi+1,τ ′ T2 . The i = 0 case of the equality (c) is intere... |

69 |
Provability of the pigeonhole principle and the existence of infinitely many primes
- Paris, Wilkie, et al.
- 1988
(Show Context)
Citation Context ...ierarchy question is independent of S2. The theory S2 can formulate facts about the density of primes, variants of Ramsey’s theorem, and can formalise many arguments used to show circuit lower bounds =-=[25,22,24]-=-. So this would be a non-trivial independence result. Nevertheless, it should be easier to separate bounded arithmetic theories than to separate the polynomial hierarchy. This is because bounded arith... |

62 |
Bounded arithmetic and the polynomial hierarchy
- Kraj́ıček, Pudlák, et al.
- 1991
(Show Context)
Citation Context ...w independence of questions like P = NP ? from some sizeable portion of mathematics. It is known that if the bounded arithmetic hierarchy S2 = ∪iS i 2 collapses, then so does the polynomial hierarchy =-=[19,9]-=-. However, it is unknown what does the failure of the bounded arithmetic hierarchy to collapse imply about the polynomial hierarchy. At our present state of knowledge, the noncollapse of the bounded a... |

60 | Notes on polynomially bounded arithmetic
- Zambella
- 1996
(Show Context)
Citation Context ...ne unbounded iterm then P H = B(Σ p i+2). It was known from Krajíček, Pudlak, and Takeuti [19] that if T i 2 = S i+1 2 the polynomial hierarchy collapses to the (i + 2)nd level. Buss [9] and Zambella =-=[29]-=- showed that if T i 2 = S i+1 2 then T i 2 proves the polynomial hierarchy collapses to the (i + 3)rd level. Both of these results make use of Herbrand’s theorem and some combinatorics; whereas, our r... |

58 | An application of boolean complexity to separation problems in bounded arithmetic
- Buss, Kraj¶·cek
- 1994
(Show Context)
Citation Context ... R− q for i odd. With probability at least two-thirds the circuit (P ℓ,k i ) ρ is P ℓ,k i−1 after a suitable renaming of variables. PROOF. We sketch this following Krajíček [18] and Buss and Krajicek =-=[6]-=-. (i) The proof of this is the same as Lemma 10.4.7 (i) in Krajíček [18]. 57s(ii) There are 2 ||ℓ(k)||2 · ki−2 different subcircuits of depth 2 in P ℓ,k i . By our assumption 2 ||ℓ(k)||2 ≤ k, and (i),... |

49 | On Truth-Table Reducibility to SAT
- Buss, Hay
- 1991
(Show Context)
Citation Context ...(w, x)) = 1 ∧ C(x, w, v)) and let B(x, v + 1) be a ˆ Σ b i-formula provably equivalent to ✷ (∃v ′ ≤ ℓ(s(x)))(∃w ′ ≤ t)(v ′ ≥ v + 1 ∧ C(x, w ′ , v ′ )). Corollary 50 is similar to a result of Buss-Hay =-=[5]-=- where they show predicates in P Σp i (log) can be written in the form (∃v ≤ |s(x)|)(A(x, v)∧¬B(x, v)) where A and B are Σb i. Our result shows the ˆ ∆b i+1-predicates of Si 2 which are also P Σp i (l... |

44 | Bounded arithmetic and lower bounds in Boolean complexity
- Razborov
- 1995
(Show Context)
Citation Context ...ierarchy question is independent of S2. The theory S2 can formulate facts about the density of primes, variants of Ramsey’s theorem, and can formalise many arguments used to show circuit lower bounds =-=[25,22,24]-=-. So this would be a non-trivial independence result. Nevertheless, it should be easier to separate bounded arithmetic theories than to separate the polynomial hierarchy. This is because bounded arith... |

31 | Two queries
- Buhrman, Fortnow
- 1996
(Show Context)
Citation Context ...s like P Σp i [k] to denote at most k queries to a Σ p i -oracle and use parentheses such as P Σp i (k) to mean O(k) queries. From Hemaspaadra, Hemaspaadra, and Hempel [14,15] and Buhrman and Fortnow =-=[4]-=- it is known that P Σp i [k] = P Σp i [k + 1] implies P H = B(Σ p i+2). Here k is a fixed number. Let ℓ be a nondecreasing, unbounded iterm. We will show that the class P Σp i ({|ℓ|}) has complete pro... |

31 |
RSUV isomorphisms
- Takeuti
- 1993
(Show Context)
Citation Context ...oracle separations 1991 MSC: 03F30, 68Q15 1 Introduction Three families of bounded arithmetic theories, R i 2 , S i 2, and T i 2, were developed in Buss [7], Allen [1],Clote-Takeuti [11], and Takeuti =-=[28]-=-. These theories Preprint submitted to Elsevier Preprint 20 February 2003shave been studied because of their close connection to computational complexity. It is known that the Σb i-definable functions... |

30 | Axiomatizations and conservation results for fragments of bounded arithmetic
- Buss
- 1990
(Show Context)
Citation Context ...x) = y and T ⊢ (∀x)(∃y ≤ t)B. In addition, A(x, w, 0) must be provable equivalent to a ˆ Π b 0-formula in T and T must prove (∃w ≤ t)A(x, w, 0). The formula B is called a Q i,τ -definition of f. Buss =-=[8]-=- gives a variant of Qi,τ-definition. The formula B(x, y) in the above definition is equivalent to a ˆ Σb i+1-formula in ˆ T i,τ 2 so if f is Qi, ˙τ -defined in ˆT i,τ 2 , it will also be ˆ Σb i+1-defi... |

30 | Relating the bounded arithmetic and the polynomial time hierarchies
- Buss
- 1995
(Show Context)
Citation Context ...ntains at least one unbounded iterm then P H = B(Σ p i+2). It was known from Krajíček, Pudlak, and Takeuti [19] that if T i 2 = S i+1 2 the polynomial hierarchy collapses to the (i + 2)nd level. Buss =-=[9]-=- and Zambella [29] showed that if T i 2 = S i+1 2 then T i 2 proves the polynomial hierarchy collapses to the (i + 3)rd level. Both of these results make use of Herbrand’s theorem and some combinatori... |

26 |
Fragments of bounded arithmetic and bounded query classes
- Kraj́ıček
- 1993
(Show Context)
Citation Context ...polynomial time with access to a Σ p i−1-oracle [7]. The Σb 1-definable functions of R1 2 are the circuit class F NC. It is also known for i > 1 that Si 2 is Σb i-conservative over T i−1 2 . Krajíček =-=[17]-=- shows that the Σb i-definable multifunctions of S i−1 2 are F P Σp i−1(wit, log), those multifunctions computed by Turing machines running in polynomial time with only logarithmically many queries to... |

26 | Lower bounds for propositional proofs and independence results in bounded arithmetic
- Razborov
- 1996
(Show Context)
Citation Context ...pendence results in bounded arithmetic. Some independence results have already been obtained using interpolation methods. Perhaps the most cleanly stated of these is Widgersen’s corollary to Razborov =-=[26]-=-: S 2 2(α) does not prove the existence of pseudo-random number generators. Although it is probably difficult to separate the bounded arithmetic hierarchy, one might ask whether there is a relativised... |

19 | Provably total functions in bounded arithmetic theoriesÊ�,Í�andÎ
- Buss, Krajicek, et al.
- 1993
(Show Context)
Citation Context ...er motivating question was whether R i 2 is Σ b i-conservative over S i−1 2 . This is a reasonable conjecture since S i 2 is Σ b i-conservative over T i−1 2 from Buss [8]. Buss, Krajíček, and Takeuti =-=[10]-=- were not able to solve this problem but did show that if the theories had a slightly faster growth rate function #3 in the language then the result held. This paper takes up this question in the pren... |

19 |
Ramsey’s Theorem in Bounded Arithmetic
- Pudlák
- 1991
(Show Context)
Citation Context ...ierarchy question is independent of S2. The theory S2 can formulate facts about the density of primes, variants of Ramsey’s theorem, and can formalise many arguments used to show circuit lower bounds =-=[25,22,24]-=-. So this would be a non-trivial independence result. Nevertheless, it should be easier to separate bounded arithmetic theories than to separate the polynomial hierarchy. This is because bounded arith... |

18 |
Arithmetizing uniform NC
- Allen
- 1989
(Show Context)
Citation Context ...ed functions, conservation results, oracle separations 1991 MSC: 03F30, 68Q15 1 Introduction Three families of bounded arithmetic theories, R i 2 , S i 2, and T i 2, were developed in Buss [7], Allen =-=[1]-=-,Clote-Takeuti [11], and Takeuti [28]. These theories Preprint submitted to Elsevier Preprint 20 February 2003shave been studied because of their close connection to computational complexity. It is kn... |

10 |
Arithmetic Theories with Prenex Normal Form Induction
- Pollett
- 1997
(Show Context)
Citation Context ... t ∗ (ℓ(s), a). Buss [7] has shown that one gets the same theory if one formulates Si 2 or T i 2 with induction axioms or induction rules. The same sorts of proof work in the ˆT i,τ 2 and Ĉi,τ 2 case =-=[23]-=-. Given a set Ψ of prenex formulas let LΨ be the formulas which can be made into Ψ-formulas by padding the left hand side with zero or more dummy quantifiers. The next result is the primary reason why... |

9 |
Separating fragments of bounded arithmetic
- Beckmann
- 1996
(Show Context)
Citation Context ...p i (X) ({||ℓ|| 2 }) where ℓ is a nondecreasing, unbounded iterm. This result implies many oracle separations. Some of these results were obtained independently by Arnold Beckmann in his Ph.D. thesis =-=[2]-=- using a technique called “dynamic ordinal analysis”. Lastly, we give a result concerning models separating theories. 7.1 Hierarchy collapses In this subsection, we use brackets in expressions like P ... |

9 | A downward translation in the polynomial hierarchy
- Hemaspaandra, Hemaspaandra, et al.
- 1997
(Show Context)
Citation Context ...n, we use brackets in expressions like P Σp i [k] to denote at most k queries to a Σ p i -oracle and use parentheses such as P Σp i (k) to mean O(k) queries. From Hemaspaadra, Hemaspaadra, and Hempel =-=[14,15]-=- and Buhrman and Fortnow [4] it is known that P Σp i [k] = P Σp i [k + 1] implies P H = B(Σ p i+2). Here k is a fixed number. Let ℓ be a nondecreasing, unbounded iterm. We will show that the class P Σ... |

8 | On the cutting edge of relativization: The resource bounded injury method
- Buhrman, Torenvliet
- 1994
(Show Context)
Citation Context ...racle X such that for all i ≥ 1 there is an ℓ for which (N, X) separates ˆ T i,{ℓ} 2 (α) from ˆ T i,{|ℓ|} 2 (α) for ˆ ∆b 2(α)-predicates yet (N, X) |= P H(α) = ∆ p 2(α). PROOF. Buhrman and Torenvliet =-=[3]-=- give an oracle X for which NEXP X ⊆ NP X P . So (N, X) |= NEXP (α) = P H(α) = P NP (α). Mocas [20] shows P NP (nk ) � NEXP and this relativizes. Now consider ˆ T i,{|id|i} 2 (α) versus ˆT i,{|id|i−1}... |

6 | Bounded arithmetic for NC, ALogTIME - Clote, Takeuti - 1992 |

6 | Translating equality downwards
- Hemaspaandra, Hemaspaandra, et al.
- 1997
(Show Context)
Citation Context ...n, we use brackets in expressions like P Σp i [k] to denote at most k queries to a Σ p i -oracle and use parentheses such as P Σp i (k) to mean O(k) queries. From Hemaspaadra, Hemaspaadra, and Hempel =-=[14,15]-=- and Buhrman and Fortnow [4] it is known that P Σp i [k] = P Σp i [k + 1] implies P H = B(Σ p i+2). Here k is a fixed number. Let ℓ be a nondecreasing, unbounded iterm. We will show that the class P Σ... |

5 |
Separating Exponential time classes from polynomial time classes
- Mocas
- 1993
(Show Context)
Citation Context ...i,{|ℓ|} 2 (α) for ˆ ∆b 2(α)-predicates yet (N, X) |= P H(α) = ∆ p 2(α). PROOF. Buhrman and Torenvliet [3] give an oracle X for which NEXP X ⊆ NP X P . So (N, X) |= NEXP (α) = P H(α) = P NP (α). Mocas =-=[20]-=- shows P NP (nk ) � NEXP and this relativizes. Now consider ˆ T i,{|id|i} 2 (α) versus ˆT i,{|id|i−1} 2 (α). For all i ≥ 1, ˆ T i,{|id|i−1} 2 (α) ⊇ T 1 2 (α) so its ˆ ∆b 2(α)-predicates contain P NP (... |